Thursday, February 23, 2012

The Hilbert program aims to formalize (axiomatize) all mathematics on a secure core foundation of “finitistic” (excluding infinite sets, etc) mathematics whose “truth” is beyond all doubt.

Gödel's incompleteness theorems prove that any such formalization that is powerful enough to embed (encode) Peano arithmetic is incomplete (i.e. there is a statement of Peano arithmetic that can neither be proved nor disproved). This shows the impossibility of the Hilbert program, and is said to have induced a “crisis” in the foundation of mathematics.Despite the occasional sensational claim that Gödel's incompleteness theorems in effect destroyed both the mathematical enterprise, and society’s and scientists’ faith in mathematics, mathematics has in fact progressed apace since 1931.

Other than logicians and theoretical computer scientists, mathematicians are, in general, blissfully uneducated in Gödel's incompleteness theorems. An algebraist or a topologist would happily pursue her/his research without any regard to the foundational “crisis“. Far from being disillusioned, mathematicians are as motivated and enthusiastic as ever.

Why?

Case 1If one is trying to prove or diaprove a statement M in a theory T (such as set theory), that is powerful enough to embed Peano arithmetic, then, as T is incomplete (as proved by Gödel), there are statements S in T such that both S and (not S) are not provable. M may be just such a statement. If so, trying to prove or disprove M is doomed to fail.

This then is the extent of one’s psychological “crisis” induced by the foundational “crisis“. However, even if the theory were complete, so that either M or (not M) is provable, proving either may well be beyond one‘s intellect. I would contend that knowing that the theory is complete does not make one's attempt to prove either M on (not M) any easier or reassuring.